What does it mean if a question asks me to find the set of "rationalizable actions" for a given game?
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$\begingroup$ See: en.wikipedia.org/wiki/Rationalizability $\endgroup$– Walrasian AuctioneerCommented Dec 7, 2020 at 13:42
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2 Answers
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A strategy is rationalizable if it survives an iterated elimination of strictly dominated strategies. That is first you eliminate all strategies that are never a best response independent what action the other player chooses. Then you only consider the strategies that survive and repeat this step: eliminate all strategies that are never a best response independent what action from the set of surviving strategies the other player chooses.
The set of rationalizable strategies is then a set that only contains best responses to some rationalizable strategy. Therefore this is a larger set than Nash equilibria, which are best responses against each other.
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$\begingroup$ Hi, is there such a thing as a rationalizable mixed strategy? Suppose I have a 4x4 game. Suppose I am not given any probabilities. Should I assign arbitrary probabilities to each strategy? $\endgroup$ Commented Dec 12, 2020 at 20:38
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The preceding answer is correct only for two-player games. In general the set of rationalizable strategies is a subset of the iteratively undominated strategies because in games with three or more players there can be strategies for one player which are best replies (and thus undominated) only to beliefs about the play of others in which play is correlated. The assumption that players choose independently requires that such undominated strategies also be eliminated.
